A298537 Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.

1, 1, 2, 3, 6, 10, 25, 49, 127, 291, 766, 1843, 5003, 12487, 34151, 87983, 242088, 634848, 1763749, 4688677, 13085621, 35241441, 98752586, 268282856, 755353825, 2067175933, 5837592853, 16087674276, 45550942142, 126186554309, 358344530763, 997171512999

Offset: 1

Author

Gus Wiseman, Jan 20 2018

Keywords

• nonn

Examples

			The a(5) = 6 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), ((o)(o)), (oooo).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000081, A003238, A004111, A032305, A289078, A289079, A290689, A291443, A297791, A298422, A298533, A298535, A298538, A298539.

Programs

Mathematica

r[n_]:=r[n]=If[n===1,1,Sum[Product[Binomial[r[x]+Count[ptn,x]-1,Count[ptn,x]],{x,Union[ptn]}],{ptn,IntegerPartitions[n-1]}]];
Table[If[n===1,1,Sum[Binomial[r[(n-1)/d]+d-1,d],{d,Divisors[n-1]}]],{n,40}]

Formula

a(n + 1) = Sum_{d|n} binomial(A000081(n/d) + d - 1, d).